Problem: Find $ \lim_{x\to 3}f(x)$ for $f(x)=\dfrac{x^2-4}{11-2x}$.
Solution: $f$ is a rational function. Rational functions are continuous across their entire domain, and their domain is all real $x$ -values that don't make the denominator equal to zero. In other words, for any rational function $r$ and any input $c$ in the domain of $r$, we know that this equality holds: $\lim_{x\to c}r(x)=r(c)$ The input $x=3$ is within the domain of $f$. Therefore, in order to find $ \lim_{x\to 3}f(x)$, we can simply evaluate $f$ at $x=3$. $\begin{aligned} &\phantom{=}f(x) \\\\ &=\dfrac{x^2-4}{11-2x} \\\\ &=\dfrac{(3)^2-4}{11-2(3)} \gray{\text{Substitute }x=3} \\\\ &=\dfrac{5}{5} \\\\ &=1 \end{aligned}$ In conclusion, $ \lim_{x\to 3}f(x)=1$.